# Full text of "Treatise On Analysis Vol-Ii"

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```9   THE PLANCHEREL-GODEMENT THEOREM       385

space H is a mapping x H-» U(x) of F into <&(H) such that U(e) = 1H , U(xy) = U(x)U(y)

and £/(**)=</(*)*.

Let E be a Hilbert space and x H* T(x) a mapping of F into «5f(E). In order that
there should exist a Hilbert space H which is the Hilbert sum of E and another Hilbert
space F, and a representation x h-+ U(x) of F on H such that T(x) = PU(x) \ E for all
x e F, where P is the orthogonal projection of H on E, it is necessary and sufficient that
T should satisfy the following three conditions :

(1)   rO?)=lE and r(x*) === T(*)* for all*eE.

(2)   If g : r -»• E is any mapping such that g(x) = 0 for all but a finite set of ele-
ments x of F, then

E     (T(x*y) •

(3) If g : F ~> E is any mapping such that g(x) = 0 for all but a finite set of elements
x e F, then for each z £ F there exists a constant M2 > 0 such that

£      (T(x*z*zy)-g(y)\g(x)-)^UI-     £     (T(x*y) • g(y) \ g(x)).

(x. jOeFxr                                                     <x, jOeFxr

Moreover, if U and H satisfy these conditions and are such that the elements
U(x) 'f(x e F and /e H) form a total set in H, then the representation U is deter-
mined up to equivalence (equivalence of representations being defined as in (15.5)).

(To show that these conditions are sufficient, let G be the subspace of Er consisting
of mappings g : F -> E such that g(x) = 0 for all but a finite set of elements x e F,
and consider the form on G x G

(g,K)^V(g,K) =      £      (T(x*y)-g(y)\h(x))

(x, y) e r x r

which is a positive hermitian form. If N is the set of all g e G such that B(#, g} ~ 0,
then B induces a nondegenerate positive hermitian form (g, h)\-+(g\ h) = B(#, A) on
G/N which makes G/N a prehilbert space. Assume that G/N is a dense subspace of a
Hilbert space H0 . Define an injection j : G/N ->• Er as follows : for each g e G, the
image/(^) of g is the mapping

By transporting the scalar product on G/N by means of y, we have a structure of pre-
hilbert space on /(G/N) and hence, extending by continuity, an isomorphism of H0
onto a Hilbert space H. Then define U(x) by the condition that U(x) *j(g) is the mapping

Let A be a separable commutative Banach algebra with involution, having a unit
element e. Let P be the set of positive linear forms on A (or of traces on A), which is a
subset of the dual A' of the Banach space A (15.6.11). If /i, /2 e P, we write/! <^/2 to
mean that /2 — /i e P.

(a)    If /,/0 are traces on A such that/0 ^/, show that there exists a sequence (^n) of
elements of A such that/0(;v) = lim/Otfx) for all x e A. (If g is the bitrace correspond-

n-+oo

ing to /, note that /0 may be written in the form /0 = u ° irg , where u is a continuous
linear form on H^ .)

(b)    Let P! be the set of traces /e P such that ||/|| =/(e) g 1 ; then Pa is convex and
weakly compact. Show that the set of extremal points of PI (Section 12.15, Problem 5)
is equal to (P n X(A)) u {0}. (Use (a) to show that if a character is a trace it mustsuch that b2 =
```